Inequality Questions and Answers

Answer: B

Correct answer: only conclusion II is true.

This set’s code reads \(\#\) as “smaller than”, \(©\) as “greater than”, \(\pi\) as \(\le\), \(\$\) as \(\ge\), and \(\%\) as “neither smaller nor greater”, that is \(=\). The statement becomes \[ L \ge M = Q > S < T \le P. \] S is a low point, so the chain reverses there.

II. \( L > S \): both elements sit on the left of the pivot, joined by \( L \ge M = Q > S \), which contains a strict link. True.

I. \( T > L \): T lies on the right of the pivot and L on the left, so no chain runs between them. With \( S = 1 \), \( L = 9, T = 2 \) and \( L = 2, T = 9 \) both satisfy every statement. Not determined.

Only II follows.

Answer: B

Correct answer: both conclusions I and II are true.

This set’s code reads \(\#\) as “smaller than”, \(©\) as “greater than”, \(\pi\) as \(\le\), \(\$\) as \(\ge\), and \(\%\) as “neither smaller nor greater”, that is \(=\). The statement becomes \[ W < H \le I = C \ge H > S. \] Note that H is named twice, which is what ties the two halves together.

I. \( C \ge W \): the run \( W < H \le I = C \) points one way throughout and opens strict, so \( C > W \) - which certainly satisfies the weaker \( C \ge W \) that the conclusion states. True.

II. \( C > S \): reading from C, \( C \ge H > S \), strict at the last link. True.

Both follow. Conclusion I is the one candidates most often reject, on the grounds that the chain gives a strict relation while the conclusion asks for a weak one; but \( \ge \) requires only that C is not smaller than W, and \( C > W \) delivers that. A strict result always satisfies the corresponding weak claim - never the reverse.

Answer: E

Correct answer: both conclusions I and II are true.

This set’s code reads \(\#\) as “smaller than”, \(©\) as “greater than”, \(\pi\) as \(\le\), \(\$\) as \(\ge\), and \(\%\) as “neither smaller nor greater”, that is \(=\). The statement becomes \[ W = X > Y \le Z > R = L. \] Y is a low point, so the chain reverses there - but neither conclusion needs to cross it.

I. \( Z > L \): both lie on the right of the pivot, and \( Z > R = L \) reads straight off. True.

II. \( Y < W \): both lie on the left, and \( W = X > Y \) gives it directly. True.

Both follow. The comparison this chain would refuse is W against Z, or W against L - anything spanning the reversal at Y - and the question asks for neither.

Answer: D

Correct answer: only conclusion I is true.

This set’s code reads \(\#\) as \(<\), \(©\) as \(>\), \(\pi\) as \(\le\), \(\$\) as \(\ge\) and \(\%\) as \(=\). The statement becomes \[ A < N = O \le T \le H < E. \] This one has no reversal at all - it climbs from A to E in a single direction - so every comparison along it can be settled, and the only question is whether the result is strict.

I. \( E \ge O \): the run \( O \le T \le H < E \) contains a strict link at the end, so in fact \( E > O \), which satisfies the weaker claim the conclusion makes. True.

II. \( H > N \): the run \( N = O \le T \le H \) is weak at every link, so it gives only \( H \ge N \). The conclusion asks for the strict relation, and equality is not excluded - taking N, O, T and H all equal satisfies every statement. Not determined.

Only I follows. The contrast between the two is the whole lesson: a chain with one strict link proves the strict conclusion and the weak one; a chain of purely weak links proves only the weak one.

Answer: A

Correct answer: only conclusion I is true.

This set’s code reads \(\#\) as \(<\), \(©\) as \(>\), \(\pi\) as \(\le\), \(\$\) as \(\ge\) and \(\%\) as \(=\). The statement becomes \[ P > V = T < D < M \ge F. \] The chain falls to T and then climbs away, so T is a low point.

I. \( M > V \): both are reached from T on the same side of nothing - \( V = T < D < M \) runs one way from V up to M, with strict links. True.

II. \( P < F \): P sits above the low point on the left, F at or below M on the right, and no chain crosses between them. With \( T = 1 \), \( M = 10 \), F may be anything from 1 upward and P anything above 1, so either order is possible. Not determined.

Only I follows.

Answer: A

Correct answer: A < E ≥ B > D > F = G < C

Reading this option, E ≥ B > D > F = G gives E > G, and G < C with G = F gives C > F. Both required relations therefore follow from unbroken chains. Each of the other options breaks at least one of them at a reversal of direction.

Answer: D

Correct answer: A > E, though the statement it depends on is missing

The statement this question refers to has been lost from the record - what survives reads only “D ≤ CE”, with a symbol missing between C and E and the earlier part of the chain gone entirely. The five options between them mention A, B, C, D, E and F, so the original statement linked at least six letters, and none of those links can be recovered from what remains. The keyed answer is \( A > E \), which tells us the original chain placed A above E; but with the chain itself absent, that cannot be demonstrated here, and the four rejected options cannot be shown to fail. The item needs its original statement restored before it can be worked.

Answer: B

Correct answer: >

The chain is C = B ? A ≥ D, and C > D needs a strict step somewhere between C and D. The link A ≥ D is not strict and C = B is not either, so the strictness must be supplied at the question mark: B > A gives C = B > A ≥ D and therefore C > D.

Answer: C

Correct answer: D,B,A,C,E

Filling the blanks gives D < B ≤ A = C > E, from which D < B ≤ A = C yields D < C. The requirement is that D sits to the left of the strict less-than and C to the right of the equals, so the chain runs upward from D to C without a reversal.

Answer: A

Correct answer: A>E

The chain A ≥ B = C > D = E runs without reversal and contains one strict step at C > D, so A > E follows. The other options fail at the reversal E ≤ F: nothing above E can be compared with F or G, so neither B < F nor C > G can be established.